Question

Consider a Poisson random variable with \(\mu=2.5 .\) Use the Poisson formula to calculate the following probabilities: a. \(P(x=0)\) b. \(P(x=1)\) c. \(P(x=2)\) d. \(P(x \leq 2)\)

Short answer

Question: Calculate the probabilities associated with a Poisson random variable with a mean of 2.5 for the following scenarios: a) x=0, b) x=1, c) x=2, d) x≤2. Answer: Using the Poisson distribution formula, we found the probabilities for each specified value: a. P(x=0) ≈ 0.0821 b. P(x=1) ≈ 0.2052 c. P(x=2) ≈ 0.2565 d. P(x≤2) ≈ 0.5438

Step-by-step solution

The Poisson distribution formula is given by \(P(X=k) =\frac{\mu^k e^{-\mu}}{k!}\), where k is the number of occurrences, \(\mu\) is the mean, and \(e\) is the Euler's number. #Step 2: Plug in the given mean value and calculate the probabilities for x=0,1, and 2#

Using the formula and the given mean \(\mu=2.5\), we will find the probabilities for x=0, 1, and 2 as follows: a. \(P(x=0) = \frac{2.5^0 e^{-2.5}}{0!}\) b. \(P(x=1) = \frac{2.5^1 e^{-2.5}}{1!}\) c. \(P(x=2) = \frac{2.5^2 e^{-2.5}}{2!}\) #Step 3: Calculate the probabilities for x=0, 1, and 2 using the Poisson formula#

By calculating the equations we have from Step 2, we get: a. \(P(x=0) = \frac{1 e^{-2.5}}{1} \approx 0.0821\) b. \(P(x=1) = \frac{2.5 e^{-2.5}}{1} \approx 0.2052\) c. \(P(x=2) = \frac{6.25 e^{-2.5}}{2} \approx 0.2565\) #Step 4: Calculate the probability of x being less than or equal to 2#

To find the probability of \(P(x \leq 2)\), we will sum the probabilities for x=0, 1, and 2: \(P(x \leq 2) = P(x=0) + P(x=1) + P(x=2) \approx 0.0821 + 0.2052 + 0.2565 \approx 0.5438\) The probabilities for each of the specified values are: a. \(P(x=0) \approx 0.0821\) b. \(P(x=1) \approx 0.2052\) c. \(P(x=2) \approx 0.2565\) d. \(P(x \leq 2) \approx 0.5438\)

Key concepts

Probability Calculation

Probability calculation in the context of the Poisson distribution is a fascinating topic. To understand this, we use the Poisson formula, which helps to find the likelihood of a given number of events occurring within a fixed interval of time or space.
It is expressed as:\[P(X=k) = \frac{\mu^k e^{-\mu}}{k!}\] where:

• \(P(X=k)\) is the probability of observing \(k\) events,
• \(\mu\) is the average number of events, known as the statistical mean,
• \(e\) is Euler's number, approximately equal to 2.71828,
• \(k!\) is the factorial of \(k\).

The key to calculating probabilities in a Poisson distribution is substituting the values of \(\mu\) and \(k\) into the formula and performing the computations step-by-step. This process uncovers the chance of specific outcomes, which in our exercise is finding probabilities for \(k=0, 1, 2\), and for cumulative values up to 2.
The Poisson probability values calculated show us how frequent certain events might be, given the average rate of occurrence \(\mu\). Calculating these probabilities provides insights into patterns and expectations in statistically random events.

Random Variables

A random variable is a fundamental concept in probability and statistics. It is a variable that represents possible outcomes of a statistical experiment. There are two main types of random variables: discrete and continuous.

In our case, the Poisson random variable we are dealing with is discrete. Discrete random variables take on specific values, often representing countable outcomes. The possible values in our exercise are 0, 1, 2, etc.
Each of these values signifies a count of events happening in a fixed period, like the number of emails received in an hour or the number of calls at a help center.
This transitioning between the mathematical world of numbers and the real-world scenarios is pivotal. It helps in applying mathematical theories to practical situations. For instance, with a Poisson random variable like this, businesses can predict the number of incoming support calls or website traffic at different times, making it a powerful tool for planning and decision-making. This blend of theory and application makes understanding random variables both exciting and essential.

Statistical Mean

The statistical mean in the context of a Poisson distribution is represented by \(\mu\), which is the average rate at which events occur. Understanding the mean is crucial as it hints at what is common or typical in a dataset.
For example, with \(\mu=2.5\) in our exercise, the mean suggests that, on average, 2.5 events occur in a given time frame. This average is a central figure that frequency calculations pivot around.

Why is the mean important? Because when we understand the mean, we can predict the expected frequency and make well-informed decisions. Knowing these expected values helps businesses and researchers anticipate demand, allocate resources efficiently, and evaluate performance.
In the realm of Poisson distribution and statistics, grasping the statistical mean provides a firm footing to understand the likelihood of events and guides the calculation of other probabilities. It serves not just as an abstract number, but a practical indicator of what is expected to happen over time.